Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1859,4,Mod(1,1859)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1859, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1859.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1859 = 11 \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1859.a (trivial) |
Newform invariants
comment: select newform
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
| Self dual: | yes |
| Analytic conductor: | \(109.684550701\) |
| Analytic rank: | \(0\) |
| Dimension: | \(11\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{11} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{11} - 5 x^{10} - 64 x^{9} + 268 x^{8} + 1564 x^{7} - 4963 x^{6} - 16942 x^{5} + 37082 x^{4} + \cdots + 16256 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 143) |
| Fricke sign: | \(1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{10}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{11} - 5 x^{10} - 64 x^{9} + 268 x^{8} + 1564 x^{7} - 4963 x^{6} - 16942 x^{5} + 37082 x^{4} + \cdots + 16256 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 13 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 73921 \nu^{10} + 8439311 \nu^{9} + 2434932 \nu^{8} - 603336064 \nu^{7} - 474655656 \nu^{6} + \cdots + 47192465320 ) / 601708456 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 130026 \nu^{10} - 1931719 \nu^{9} - 3195319 \nu^{8} + 103863653 \nu^{7} + 20780639 \nu^{6} + \cdots - 4999000508 ) / 150427114 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 833811 \nu^{10} + 2410905 \nu^{9} + 50444340 \nu^{8} - 75562068 \nu^{7} - 1065164652 \nu^{6} + \cdots - 1290598552 ) / 601708456 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 499260 \nu^{10} - 759762 \nu^{9} + 34818189 \nu^{8} + 92778516 \nu^{7} - 721641868 \nu^{6} + \cdots - 8760781816 ) / 150427114 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 2635821 \nu^{10} - 5571119 \nu^{9} - 174619828 \nu^{8} + 157660112 \nu^{7} + 3998941440 \nu^{6} + \cdots - 11305102248 ) / 601708456 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 3965935 \nu^{10} + 6054403 \nu^{9} + 281696508 \nu^{8} - 144928752 \nu^{7} + \cdots + 35528598512 ) / 601708456 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 6914567 \nu^{10} - 19945349 \nu^{9} - 450124744 \nu^{8} + 773726616 \nu^{7} + 10426407568 \nu^{6} + \cdots - 44225190288 ) / 601708456 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 7435567 \nu^{10} + 14036427 \nu^{9} + 506760676 \nu^{8} - 378150932 \nu^{7} + \cdots + 59984105152 ) / 601708456 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 13 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{10} - \beta_{8} + \beta_{7} - \beta_{5} + 3\beta_{2} + 23\beta _1 + 15 \)
|
| \(\nu^{4}\) | \(=\) |
\( 4 \beta_{10} + 2 \beta_{9} - 2 \beta_{8} + 5 \beta_{7} + 3 \beta_{6} - 3 \beta_{5} - 3 \beta_{4} + \cdots + 288 \)
|
| \(\nu^{5}\) | \(=\) |
\( 45 \beta_{10} + 9 \beta_{9} - 39 \beta_{8} + 45 \beta_{7} + 9 \beta_{6} - 25 \beta_{5} - 6 \beta_{4} + \cdots + 774 \)
|
| \(\nu^{6}\) | \(=\) |
\( 214 \beta_{10} + 116 \beta_{9} - 122 \beta_{8} + 224 \beta_{7} + 143 \beta_{6} - 91 \beta_{5} + \cdots + 7758 \)
|
| \(\nu^{7}\) | \(=\) |
\( 1716 \beta_{10} + 657 \beta_{9} - 1316 \beta_{8} + 1541 \beta_{7} + 577 \beta_{6} - 422 \beta_{5} + \cdots + 31051 \)
|
| \(\nu^{8}\) | \(=\) |
\( 9061 \beta_{10} + 5572 \beta_{9} - 5389 \beta_{8} + 7817 \beta_{7} + 5650 \beta_{6} - 1185 \beta_{5} + \cdots + 237000 \)
|
| \(\nu^{9}\) | \(=\) |
\( 63313 \beta_{10} + 34338 \beta_{9} - 44023 \beta_{8} + 48504 \beta_{7} + 27533 \beta_{6} + \cdots + 1155998 \)
|
| \(\nu^{10}\) | \(=\) |
\( 356210 \beta_{10} + 249413 \beta_{9} - 212170 \beta_{8} + 250295 \beta_{7} + 216294 \beta_{6} + \cdots + 7773705 \)
|
Embeddings
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
comment: embeddings in the coefficient field
gp: mfembed(f)
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−5.35762 | 7.47772 | 20.7041 | 10.6979 | −40.0628 | 32.3181 | −68.0635 | 28.9163 | −57.3151 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −5.05967 | −10.2152 | 17.6003 | 4.34036 | 51.6854 | −31.2716 | −48.5744 | 77.3494 | −21.9608 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −4.37601 | 4.08546 | 11.1495 | −1.30050 | −17.8780 | −14.9266 | −13.7821 | −10.3090 | 5.69101 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −4.16556 | −0.469632 | 9.35186 | −20.3915 | 1.95628 | −2.83667 | −5.63125 | −26.7794 | 84.9419 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.39026 | 8.98097 | −6.06718 | −6.95692 | −12.4859 | −9.63375 | 19.5570 | 53.6578 | 9.67193 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −0.363322 | −3.74669 | −7.86800 | 11.8947 | 1.36126 | −28.3874 | 5.76520 | −12.9623 | −4.32160 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −0.221588 | −7.13795 | −7.95090 | −6.37060 | 1.58168 | 23.0480 | 3.53453 | 23.9503 | 1.41165 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.58491 | −2.54183 | −1.31822 | 20.1048 | −6.57040 | 29.2165 | −24.0868 | −20.5391 | 51.9693 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 2.59635 | 4.20666 | −1.25897 | −11.3514 | 10.9220 | −3.36670 | −24.0395 | −9.30403 | −29.4723 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 4.65354 | 8.62383 | 13.6554 | 21.5680 | 40.1313 | −9.18213 | 26.3178 | 47.3704 | 100.367 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 5.09923 | −3.26338 | 18.0021 | −18.2348 | −16.6407 | −29.9777 | 51.0031 | −16.3503 | −92.9834 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(11\) | \(1\) |
| \(13\) | \(1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1859.4.a.e | 11 | |
| 13.b | even | 2 | 1 | 143.4.a.d | ✓ | 11 | |
| 39.d | odd | 2 | 1 | 1287.4.a.m | 11 | ||
| 52.b | odd | 2 | 1 | 2288.4.a.u | 11 | ||
| 143.d | odd | 2 | 1 | 1573.4.a.f | 11 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 143.4.a.d | ✓ | 11 | 13.b | even | 2 | 1 | |
| 1287.4.a.m | 11 | 39.d | odd | 2 | 1 | ||
| 1573.4.a.f | 11 | 143.d | odd | 2 | 1 | ||
| 1859.4.a.e | 11 | 1.a | even | 1 | 1 | trivial | |
| 2288.4.a.u | 11 | 52.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{11} + 6 T_{2}^{10} - 59 T_{2}^{9} - 368 T_{2}^{8} + 1134 T_{2}^{7} + 7525 T_{2}^{6} - 7730 T_{2}^{5} + \cdots + 8808 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1859))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{11} + 6 T^{10} + \cdots + 8808 \)
|
| $3$ |
\( T^{11} - 6 T^{10} + \cdots - 10592776 \)
|
| $5$ |
\( T^{11} + \cdots - 58263405696 \)
|
| $7$ |
\( T^{11} + \cdots - 7302898028448 \)
|
| $11$ |
\( (T + 11)^{11} \)
|
| $13$ |
\( T^{11} \)
|
| $17$ |
\( T^{11} + \cdots - 24\!\cdots\!52 \)
|
| $19$ |
\( T^{11} + \cdots + 21\!\cdots\!76 \)
|
| $23$ |
\( T^{11} + \cdots + 72\!\cdots\!84 \)
|
| $29$ |
\( T^{11} + \cdots - 37\!\cdots\!28 \)
|
| $31$ |
\( T^{11} + \cdots + 39\!\cdots\!84 \)
|
| $37$ |
\( T^{11} + \cdots + 21\!\cdots\!32 \)
|
| $41$ |
\( T^{11} + \cdots + 61\!\cdots\!72 \)
|
| $43$ |
\( T^{11} + \cdots - 91\!\cdots\!44 \)
|
| $47$ |
\( T^{11} + \cdots + 11\!\cdots\!76 \)
|
| $53$ |
\( T^{11} + \cdots - 58\!\cdots\!96 \)
|
| $59$ |
\( T^{11} + \cdots - 13\!\cdots\!24 \)
|
| $61$ |
\( T^{11} + \cdots - 15\!\cdots\!16 \)
|
| $67$ |
\( T^{11} + \cdots + 30\!\cdots\!12 \)
|
| $71$ |
\( T^{11} + \cdots + 34\!\cdots\!28 \)
|
| $73$ |
\( T^{11} + \cdots - 21\!\cdots\!04 \)
|
| $79$ |
\( T^{11} + \cdots + 58\!\cdots\!08 \)
|
| $83$ |
\( T^{11} + \cdots - 53\!\cdots\!32 \)
|
| $89$ |
\( T^{11} + \cdots + 10\!\cdots\!48 \)
|
| $97$ |
\( T^{11} + \cdots + 23\!\cdots\!28 \)
|
| show more | |
| show less |